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Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Harmonic and anharmonic oscillations investigated by using a microcomputer-based Atwood’s machine Barbara Pecori Phys. Department of Bologna University, Italy Giacomo Torzoa) and Andrea Sconza Phys. Department of Padova University, Italy Received 11 February 1998; accepted 26 August 1998 We describe how the Atwood’s machine, interfaced to a personal computer through a rotary encoder, is suited for investigating harmonic and anharmonic oscillations, exploiting the buoyancy force acting on a body immersed in water. We report experimental studies of oscillators produced by driving forces of the type Fkxn with n1,2,3, and Fk sgn(x). Finally we suggest how this apparatus can be used for showing to the students a macroscopic model of interatomic forces. © 1999 American Association of Physics Teachers.
I. INTRODUCTION features of the motion of an oscillating cylinder for small amplitudes (Fkx) in Sec. II A and for large amplitudes Simple Harmonic Motion SHM is usually studied in in- (Fk sgn x) in Sec. II B; in Sec. II C we study the case of troductory physics courses because it involves easy math- an oscillating triangle (Fkxkx2) and in Sec. II D the ematics and because two simple experiments may be used as case of an oscillating cone (Fkxkx2kx3). In Sec. examples: the pendulum and the spring-mass system. Experi- III we describe the experimental setup and in Sec. IV the mental studies are often based only on measurements of the procedure used to characterize the system. In Sec. V we re- period from which the dynamic parameters of the system are port the data obtained for the different types of oscillations, derived. we compare the predictions provided by the models with the However, in real systems the linear behavior, implicit in experimental results, and we exploit the discrepancies to re- SHM, is rarely obeyed: most oscillators are only approxi- fine the models. Conclusions are drawn in Sec. VI and a mately harmonic in the small-amplitude limit while some calculation of the period versus amplitude for various restor- interesting features may only be explained if anharmonicity ing forces is reported in the Appendix. is taken into account e.g., the expansion coefficient and the specific heat of solids, jumping phenomena, transition to chaos,.... To investigate anharmonic oscillations we may study the II. OSCILLATIONS WITH THE ATWOOD’S pendulum where the restoring force is Fmg sin : by MACHINE retaining the first two terms of the series expansion sin The Atwood’s machine is a device where two masses hang 3 1 3 /6¯ ; we get Fmg( /6) from the ends of a string passing over a pulley that can freely 3. rotate on its horizontal axis. A particular feature of the pendulum is that the restoring In the apparatus used in this study one of the two masses force mimics that of a spring that softens at larger ampli- (m2 with volume V2) dips into a water bath Fig. 1. tudes. Real springs, on the contrary, become stiffer at larger The motion of the whole system can be simply described 3 amplitudes, the restoring force being F kx kx . Both by the motion of the mass m2 : here we use an orthogonal in the pendulum and in the mass-spring system the ratio reference frame with a downward directed vertical axis. between the cubic and the linear term cannot be freely var- If we assume a massless and inextensible string, the linear 1 ied: it is constant (/ 6) in one case, while it depends acceleration a of the mass m2 may be calculated by equating on the mechanical properties of the spring material in the the torque T applied to the pulley to the rate of change of its 2 other. angular momentum d(I)/dt: A simple way to explore both harmonic and anharmonic oscillations, with a free and easy choice of the system anhar- T R21dI/dt, 1 monicity, is to use a modified Atwood’s machine, where one where T R(21) is the net driving torque in the absence of the two masses hanging from the pulley drops into a water of friction, R is the pulley’s radius, I is the momentum of bath. inertia, and 1 and 2 are the tensions applied to the string by The essential feature of this setup is that, since the masses m and m , respectively. Archimedes’ force is a function of the immersed body vol- 1 2 The pulley’s angular momentum is LI, where ume, the behavior of the restoring force F(x) can be changed R relates the linear velocity of m to the pulley’s an- by modifying the body shape. 2 gular velocity in the absence of wire slipping. In this paper we report some experiments involving vari- Each one of the string’s tensions is related to the accelera- ous types of restoring forces, all of them exploiting the buoy- tion a and to the other forces applied to each body, by the ancy effect. Newton’s law: The physical model of this modified Atwood’s machine is described in Sec. II; we derive from the model the main 1m1ag and 2m2gaFAh, 2
228 Am. J. Phys. 67 3, March 1999 © 1999 American Association of Physics Teachers 228
Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Fig. 2. The displacement versus time plot for an oscillator Fk sgn(x). Fig. 1. Schematic of the Atwood’s machine.
where kr2g, showing that we are dealing with a simple where FA(h) is the Archimedes force that, being propor- harmonic motion whose period is tional to the volume of the displaced water, is a function of 2 the immersed height h. T2M/k2M/r g. 8 Relations 2 may be used to eliminate 1 and 2 from Eq. In other words, as the cylinder rises above its equilibrium 1, point, the Archimedes force decreases because a smaller wa- R Rm m gm m aF h/dt ter volume is displaced. This increases the downward force, 2 1 2 1 2 1 A and causes the cylinder to fall. If it drops below its equilib- Ia/R, 3 rium point, the Archimedes force increases because a greater volume is displaced; this causes the upward force to increase giving and the cylinder to rise. mgFAh This description of the motion is valid as long as the os- a . 4 m m I/R2 cillation amplitude does not exceed an upper limit above 1 2 which the cylinder is completely immersed or completely out If mm2m1 is less than V2 where is the water den- of the water. sity, the Archimedes force balances the gravity force mg for some value h0 of the immersed height h: FA(h0) mg, and for any displacement from the equilibrium po- B. Oscillations due to a constant restoring force: sition the restoring force F acting on the system is the op- Fk sgn X posite of the change of the buoyancy force FAFA(h) If the oscillation amplitude is made very large (Xl, mg. In fact, when the body dips positive displacement where l is the cylinder length: the cylinder spends most of the buoyancy force increases i.e., its change is positive and the time completely outside or completely inside the water the restoring force is directed upward i.e., is negative in the bath, and in each case the restoring force is constant. In fact, chosen reference frame: FFA . the restoring force when the whole cylinder is outside water 2 If we define Mm1m2I/R as the total inertial mass is mg and when the whole cylinder is inside water is of the system,3 the acceleration may be written as mgFA(l)mg(h0l)/h0. If we want to obtain a motion that is symmetric around the aFA /M. 5 equilibrium point, we must let h0l/2. In this case, if we If we want to take into account the effect of dissipative neglect the short transient during which the cylinder is par- forces, we may add to the driving torque TR(21)a tially immersed, the absolute value of the force (l/2)r2g friction torque T f RFF , where with FF we define an mgremains constant while its sign changes when the effective friction force directed opposite to 2) applied to cylinder crosses the water surface: m . In this case the acceleration a of m may be written as 2 2 Fk sgnX, 9 aaF /M, 6 F where k/(l/r)r2g and the acceleration is where the sign depends on the direction of the velocity, 2 which may be the same as or opposite to that of the accel- ak/Ml/2r g/M. 10 eration. In this type of oscillation4 the period is directly proportional A. Harmonic oscillations of a cylinder: FkX to the square root of the amplitude A, as may be easily seen by inspecting Fig. 2. The X(t) graph is made, in fact, of 2 When the body m2 is a cylinder of radius r, the parabolic branches (Xat /2), and within each branch the 2 Archimedes force is FA(h)r hg, and its change FA relation between the base t and the height A is described by 2 may be written as a function of the displacement Xhh0 the equation Aa t /2. The period is therefore 2 from the equilibrium position: FA(X)r g(h0X) 2 2 2 T4 t42A/a8M/r lgA. 11 r h0gr gX. This system is equivalent to the well-known mass-spring system, the ‘‘elastic constant’’ being determined by the cyl- C. Oscillations of twin triangles: FkXX inder radius r and by the liquid density . If we use, instead of a body with constant cross section,a From relation 5 we obtain the cylinder acceleration: body whose cross section changes linearly, the buoyancy ak/MX, 7 forces increases quadratically with depth.
229 Am. J. Phys., Vol. 67, No. 3, March 1999 Pecori, Torzo, and Sconza 229
Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Fig. 3. Schematics of the body shaped as twin triangles.
Suppose our body is a slab in the shape of twin triangles attached at their vertex and that the counterweight is chosen Fig. 4. Block diagram of the experimental setup. to keep the twin triangles in equilibrium half-submerged in the water bath. When the twin triangles are displaced by a length X from A cubic restoring force, as shown in the Appendix, pro- equilibrium, the change in the displaced water volume may duces oscillations whose period varies with amplitude as be written by V(X)sb(X)X/2sBX(tan )X, where B is the triangle base, s the thickness, the aperture angle, and T7.42M//A. 17 sb(X)2s(tan )X the cross section at the distance X from As for the case of real twin triangles, also for real twin cones, the vertex Fig. 3. one has to account for the finite cross section at the joint of 3 We are here assuming b(0)0, i.e., a zero cross section at the cones. If r0 is the radius at the joint, the factor X in 15 2 2 3 the twin triangles’ vertex, so that the restoring force F and 16 becomes 3(r0 /tan ) X3(r0 /tan )X X , pro- FA(X)gV(X) becomes ducing an asymmetric force: 2 2 2 3 FXXX sgnX, 12 F3r0 /tan X3r0 /tan X X , 15 2 2 3 and the acceleration, from relation 5: a/M3r0 /tan X3r0 /tan X X . 16 aFAX/M/MXX, 13 The asymmetry, however, is small due to the fact that r where s tan g. 0 A simple calculation, reported in the Appendix, proves r. On the contrary, when the oscillation occurs around an that a quadratic restoring force produces oscillations whose equilibrium position with the cone vertex well below the wa- period varies with amplitude as ter surface e.g., X0h/2, where h is the cone height the restoring force turns out to be strongly asymmetric. This be- T6.87M//A. 14 comes important if we use a single cone5 with amplitudes If the triangles’ vertex, in equilibrium, is placed at a distance XX0 , where the restoring force is X from the water surface, the restoring force becomes 3 3 3 2 3 0 FXX0 X03X0X3X0X X . asymmetric. It may be written as FA(XX0)(X 18 X0)X0X0, assuming X00 for vertex below the free surface at equilibrium and X00 above. III. DATA ACQUISITION SETUP Actually a real body shaped as twin triangles cannot have a zero cross section at the midpoint b(0)b ; see Fig. 3, The data acquisition system we used is based on a Serial 0 Interface6 connecting the host computer either a Macintosh and therefore, even in the case of symmetric behavior (X 0 or a PC to a position sensor. The interface and the computer 0), a linear term (b0 /tan )X in the restoring force can- communicate by the standard serial line RS-232. The not be completely avoided. logged data can be displayed on the computer monitor in various graphical representations using the dedicated soft- ware, which also allows one to perform numerical fits and 3 D. Oscillations of twin cones: FkX other data handling. A real-time visualization of the kine- matic variables can thus be obtained. If the twin triangles are replaced by twin cones with base The position sensor is made of an optical encoder, at- radius r and aperture , the Archimedes’ force has a cubic tached to the pulley shaft7 that measures the pulley’s rotation dependence on the displacement X. angle (t) from which one gets the linear displacement X(t) With a proper choice of the counterweight, the twin cones may be set in equilibrium with their vertex at the water sur- of the hanging masses as X R once known as the pulley face (X 0). In this case a simple calculation of the dis- radius R . 0 The software calculates from the measured values of the placed water volume shows that the restoring force and the acceleration become, respectively, position the corresponding values of the velocity (t) dX(t)/dt and of the acceleration a(t)d(t)/dt. 3 FX , 15 A cheap and solid stand is provided by an aluminum tube 1 in. diam, 1.5 m long and a vise clamped to the border of a/MX3, 16 a table Fig. 4. The tube is held vertically by screwing two where (tan )2g/3. metal blocks, with vertical V grooves, to the vise lips.
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Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Fig. 5. Applied force F versus measured acceleration a in the classic At- wood’s experiment. The best fit straight line gives both the friction force and the total inertial mass. Fig. 6. Plots of position, velocity, and acceleration versus time for the cyl- inder oscillating inside and outside water.
The pulley is a metal disk 3 mm thick, radius R V. EXPERIMENTAL INVESTIGATION OF 50 mm) with a thin groove to keep the string a dacron HARMONIC AND ANHARMONIC OSCILLATIONS fishing wire 0.3 mm diam, 0.125 g/m), and it is coaxi- IN WATER ally fixed to the rotary sensor. A. The case of the cylinder The masses m1 and m2 are brass bodies, attached to each end of the string by means of a screw with a thin axial hole Figure 6 shows the plots of position, velocity, and accel- the string can pass through the hole but a string knot is eration versus time, obtained with a brass cylinder weight blocked. In order to avoid unwanted lateral oscillations 9 m2143.6 g, diameter 15 mm and height about 100 mm. when the motion is started, a small electromagnet can be The counterweight (m1135.0 g) was chosen so that in used to block and release the counterweight m1 a cylinder equilibrium the cylinder is half immersed.10 In this system that bears an iron screw at its bottom. the motion is expected to obey the predictions of the models We used a perspex vessel 100 cm height and 10 cm diam described in Secs. II A at small amplitude and II B at large in order to make visible the whole m2 path, but it may well amplitude. be replaced by cheaper glass or metal vessels. After some oscillations, when the cylinder does not com- pletely exit from water for t45 s), the motion becomes damped harmonic, as clearly shown by the same plots ex- panded in Fig. 7. IV. SYSTEM CHARACTERIZATION THE CLASSIC „ The whole motion of the cylinder may be represented by ATWOOD’S MACHINE… two models: anharmonic oscillation prevailing in the initial Before performing the experimental study of the various phase (t 45 s) and harmonic oscillation in the final phase type of oscillations, one needs to characterize the system by (t45 s). measuring the pulley’s momentum of inertia I and the effec- In plots like those shown in Figs. 6 or 7, one can use the tive friction force F . mouse to move a vertical line corresponding to the same F The easiest way to measure both I/R2 and F is to per- value of the variable t along all the plots, thus detecting the F values of all the other variables. In this way one can easily form a set of measurements of the acceleration where both measure the time values t when the acceleration changes masses move in air. In this case (F 0) relation 6 be- i A comes sign (X 0). The differences, ti ti1 , give the values of the half-periods Ti/2. In the same way one can measure the 2 amgFF/I/R m1m2 or values of the maximum and minimum displacements Xi , 19 which give the amplitude Ai . mgFFaM.
That is, if the sum of the two masses (m1m2) is kept constant, a plot of different values of the driving force mg(m2m1)g versus the measured values of the accel- 2 eration has slope (I/R m1m2)M and intercept FF . This experiment may be easily performed by moving a set of extra masses of known weight from one body to the other. An example of such a plot, obtained using two equal bod- ies of mass 118.3 g and ten extra masses of 1.12 g each so 8 that m1m2247.8 g), is shown in Fig. 5. The experimental data show a linear behavior, thus con- firming that FF can be assumed to be constant. The best fit 3 gives FF(51)10 N and a slope M(0.347 0.003) kg. Therefore the effective inertial mass of the pul- 2 ley is I/R M(m1m2)(0.0990.003) kg. Fig. 7. An expanded plot of the data of Fig. 6 harmonic motion.
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Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Fig. 10. Position versus time and acceleration versus position plots for the cylinder in the harmonic phase of the motion. The linear fit is made on the selected data. Fig. 8. Period of the cylinder oscillations as a function of the square root of amplitude. Full dots: first phase of the motion. Open dots: second phase harmonic motion. ter may be obtained as the slope of a linear fit of the velocity versus the time plot Fig. 9a. We obtain as average abso- lute value of the positive and of the negative slopes a In Fig. 8 we report the values of the period as a function of (0.2270.002) m/s2, to be compared with the value pre- A: the full dots correspond to the first phase of the motion dicted by relation 10 a(0.2230.002) m/s2. Also, here the best fitting line is also drawn, while the open dots cor- the errors bars do barely overlap, owing to the excess mass respond to the second phase the expected value T of the water film, neglected in relation 10, that is respon- 2 2M/gr 2.93 s of the harmonic motion is sible for the larger observed mean acceleration. shown. Looking in more detail at the v(t) plot, which at first sight The plot shows that the period is constant for small values looks like a triangular wave, we discover that the slope is of the amplitude A, i.e., when the cylinder is partially im- slightly different in the first half and in the second half of mersed for A6 cm), and increases linearly with A at each ‘‘quasilinear’’ portion of the plot. When the cylinder is larger amplitudes. The slope predicted by relation 11: T inside water, the acceleration is larger when the cylinder is 42/aA42M/mgA is 12.00.2 assuming an dipping into the water Fig. 9b: a0.2480.001 m/s2 uncertainty of 3 g on M and of 0.1 g on m), while the and smaller when it is rising back toward the water surface value obtained by fitting the experimental data is lower Fig. 9c: a0.2090.001 m/s2. (11.70.1 s m1/2). These two values are compatible the The same happens when the cylinder is outside the water: error bars do slightly overlap but one may still suspect that here the acceleration is slightly larger when the cylinder is some second-order effect was neglected. rising toward the top position and smaller when it is falling A reduction of the measured slope with respect to the pre- back toward the water surface. dicted one cannot be explained by considering effects such The observed discontinuities a in the acceleration are as hydrodynamic mass11 extra inertial mass due to the water systematic and larger than the experimental uncertainties. displaced by the moving cylinder because it should increase This feature reminds one of the behavior of a cart riding the measured slope with respect to the predicted one. The upward and downward on an incline, and it can indeed be discrepancy may be explained by a small increase say 0.2 g justified by taking into account friction. of the mass difference m8.6 g. The oscillating cylinder During the selected portion of motion in Fig. 9b the pul- in fact, after being dipped into the water, remains wetted by ley rotates clockwise, while during the motion selected in a water layer,12 and one may even see one water drop falling Fig. 9c it rotates counterclockwise, and therefore the fric- just after it leaves the surface. tion force has the same direction of the acceleration in the A direct measurement of the acceleration of the body first case and the opposite one in the second case. We there- when it is completely inside or completely outside the wa- fore obtain the friction force for the cylinder underwater as 3 FFM(a/2)(71)10 N. When the cylinder is in 3 air we obtain FF(41)10 N, in agreement with the value obtained from the plot of Fig. 5, within the experimen- tal uncertainties.
Fig. 9. Cylinder velocity plots. a the selected area refers to motion under- water; b the selected area refers to downward motion; c the selected area Fig. 11. Plots of position, velocity, and acceleration versus time, and accel- refers to upward motion. eration versus position for the twin triangles.
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Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Fig. 12. Frequency of the twin triangles’ oscillations as a function of the Fig. 14. Frequency of the twin cones’ oscillations as a function of the square root of amplitude. Full line: purely quadratic restoring force. Dotted amplitude. Full line: a purely cubic restoring force. Dotted line numerical 3 line: numerical integration of the law of motion. integration of the law of motion for the force FA(X)X 2 2 3(r0 /tan )X3(r0 /tan ) X .
A larger frictional force for the cylinder underwater is consistent with the extra dissipation due to the water viscos- versus the square root of the amplitude A, since we expect, ity. from Eq. 13 a linear behavior for f (A), with a slope of Turning to the harmonic phase of the motion, Eq. 7 gives 0.73 Hz/m full line. a prediction of the slope of the curve a(X): k/M 2 2 A qualitative agreement with the predicted behavior is gr /M4.57 s . In Fig. 10 we report a fit of the shown indeed by the experimental results at larger ampli- central part of the plot a(X) that for X6 cm) that gives a tudes; at smaller amplitudes, however, a clear departure from slightly larger slope: (4.660.05) s 2. the x2 oscillator model is apparent. This discrepancy may be explained by releasing an im- A much better fit is obtained by taking into account also plicit assumption that we made in our simple model. If we the linear term due to the finite thickness (b02 mm) of the consider that the vessel has a relatively small inner radius twin triangles at their vertex: the dotted curve in Fig. 12 (R5 cm), we may suspect that the vertical shift of the wa- represents the values of the frequency calculated, for each ter level due to the volume of the water displaced by the amplitude, by numerically integrating relation A3, as ex- cylinder immersion cannot be neglected. If we take into ac- plained in the Appendix. count this effect,13 by replacing the variable X with the vari- able XX1r2/(R2r2)1.023X, the expected slope 2 changes into 4.67 s , in excellent agreement with the C. The case of the twin cones measured value. A record of the motion of a twin cones-shaped body two brass cones, screwed together at their vertex, base radius r B. The case of the twin triangles 1.5 cm, height h10 cm, m2494 g, counterweight m1 Figure 11 shows the plots of position, velocity, and accel- 461 g, radius at joint r00.125 cm, tan 0.1375) is eration versus time, and acceleration versus position ob- shown in Fig. 13. tained with brass twin triangles with s1 cm, h010 cm, In the graphs we measured the values of the period and of b4.8 cm, b00.2 cm, m1384.8 g, m2408.9 g), using the amplitude, and then we plotted the frequency versus the a short vessel with a large inner diameter 30 cm to make amplitude, as we expect the behavior predicted by Eq. 16: negligible the vertical shift of the water level when the body f 0.135/MA0.135tan 2g/3MA. 20 dips into the water. Measuring the period and the amplitude, as we did for the In Fig. 14 we have drawn the straight line representing the cylinder oscillations, we obtain for the twin triangles the data expected behavior for a purely cubic restoring force a full shown in Fig. 12, where we plotted the frequency f 1/T line with slope 1.83A Hz/m). The dotted curve represents the numerical integration of the law of motion see the Ap- pendix that assumes a restoring force which includes also the linear and the quadratic term predicted by Eq. 15: 3 2 2 F(X)X 3(r0 /tan )X3(r0 /tan ) X , where r0 is the radius of the cross section of the twin cones vertex.
Fig. 13. Plots of position, velocity, and acceleration versus time, and accel- eration versus position for the twin cones. Fig. 15. Behavior of the expected restoring force for a single cone.
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Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission VI. CONCLUSIONS
The experimental results reported in this paper show that a MBL version of the Atwood’s machine is a simple and pow- erful device for investigating various kind of motions. In particular, it can be used to introduce the students to the study of harmonic and anharmonic oscillations, the required physics background being restricted to elementary mechan- ics. At the introductory level, the study of nonlinear oscillators appears to be important in order to provide the students with examples of asymmetric restoring forces that are necessary to model the real behavior of solids, e.g., the nonzero ther- mal expansion coefficient. The apparatus described here combines efficiently the ad- vantages of the Atwood’s machine and those of the MBL Fig. 16. Plot of position a and restoring force b vs time for the single acquisition system. The Atwood’s machine is suitable for cone. Plot of force versus position c: the dotted line represents the fitting introductory level investigations because the basic features 3 3 function F(XX0) X0. of the system influencing the motion of the bodies can be easily identified and modified by the students themselves. The data acquisition system can be a powerful cognitive tool by allowing a real time visualization of the relevant variables selected by the students according to any particular model D. The case of a single cone: a macroscopic model of the they wish to test, and by encouraging the students to play the atomic vibration in solids game of gradually refining the schematization in order to reach a satisfactory agreement with experimental data. The The analysis made in Sec. II D shows that for a single data accuracy that can be achieved with MBL is such to cone of height h, in equilibrium at half height (X0h/2), the make the refinements possible and testable, as shown in the 3 3 present paper. restoring force becomes FA(XX0) X0, with 2 X0XX0 , and (tan ) g/3. The geometry of this system, with the expected shape of the restoring force is shown in Fig. 15. APPENDIX: THE CALCULATED DEPENDENCE OF Comparing this system to the well-known mass-spring PERIOD FROM AMPLITUDE system, we note here that the spring stiffness is no longer constant: it increases for X0 and decreases for X0. In the A simple dimensional argument that yields the correct de- pendence of the period T from the amplitude A in a generic real world these kinds of forces are much more common than oscillation driven by a restoring force of the type Fkxn one might suppose at first sight. The cohesive force in solids is the following. The period may depend only on the quan- is the simplest example. Short-range repulsion the ‘‘hard tities sphere interaction’’ due to the Pauli exclusion principle var- ies with displacement with respect to the equilibrium posi- m kg, k N/mnkg m1n s2, A m. tion of the atoms much faster than long range attraction the The only combinations of these quantities with the dimen- van der Waals interaction due to polarization. Therefore the sions of time, for different values of n, are m/k for n1, atoms’ vibration in a crystal lattice is more closely approxi- m/kA for n2 and m/kA2 for n3. Therefore the period mated by the motion of a cone-shaped body floating in a liquid bath than by the usual mass-spring system. In fact, must be proportional to m/k, m/k/A, and m/k/A, re- only the asymmetry of the restoring force and of the related spectively. The proportionality constant must obviously be derived by potential curve may explain the positive thermal expansion a different method.15 coefficient:14 a symmetric restoring force would give instead The change in kinetic energy E between v0 for x zero thermal expansion coefficient. x , and the generic (x) for xA equals the work done If we unscrew the twin cones and change the counter- 0 v by the restoring force, and therefore we may write weight in order to keep in equilibrium half immersed one 1 x single cone, the recorded oscillation is indeed asymmetric 2 Fig. 16a. The restoring force may calculated from the m x Fxdx. A1 2 x0 measured acceleration as F(t)Ma(t), knowing that the inertial mass is now 0.544 kg. The calculated values Fig. By solving this equation with respect to the velocity, we get 16b of the variable F(t) may be plotted versus the corre- x Fx dx sponding values of the variable X(t) Fig. 16c. In this plot, x2 dx or x m dt the slope corresponds to the elastic constant in the mass- 0 A2 spring system, and it is here larger for X0 than for X0. dx In the same plot the dashed line represents the values of the dt . 3 3 2 x function F(XX0) X0, calculated with the nomi- Fxdx 3 m x nal values 200 N/m and X00.05 m. 0
234 Am. J. Phys., Vol. 67, No. 3, March 1999 Pecori, Torzo, and Sconza 234
Downloaded 22 Jan 2013 to 194.254.61.41. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission The integral in the last equation from the time of maximum 3When also the string’s mass cannot be neglected, the maximum relative displacement (x A) to the time when the system is in equi- correction to the acceleration value 4 is 2y/m, where is the linear 0 density of the wire and y is the distance of each mass from the equilibrium librium (x0) gives the value T/4, and therefore the period position the acceleration being a(m2y)gFA/(2LR) T is M, where L is the string length C. T. P. Wang, ‘‘The improved determination of acceleration in the Atwood machine,’’ Am. J. Phys. 41, 0 dx T4 A3 917–919 1973. For large cross-section dipped bodies also the hydrody- namic mass m km should be taken into account: this, however, re- A 2 x h 2 Fxdx quires the calculation of the geometrical factor k that depends on the body shape. m A 4A detailed analysis of this kind of motion was made by I. R. Gatland, the minus sign in the last equation is due to the fact that ‘‘Theory of a nonharmonic oscillator,’’ Am. J. Phys. 59, 155–158 1991. when the body approaches equilibrium its velocity is nega- 5We therefore disagree with the analysis reported by A. Jafari ‘‘Experi- tive. We may use this equation to calculate the period of an mental test of Fkxnx,’’ Phys. Teach. 34, 196 1996, where he claims oscillation due to a restoring force of the kind: Fkxn, that a single cone would produce a purely cubic restoring force. 6The data and graphics reported in this paper were taken with an interface 0 dx model ULI-II produced by Vernier Software Portland, OR, but similar T4 performances were obtained using an equivalent interface model 500 A 2 x produced by PASCO Roseville, CA. kxndx 7The ‘‘Rotary motion sensor’’ is available both from Vernier Software and m A PASCO. A similar device can be home built using one of the encoders contained in the ‘‘mouse’’ that comes with any PC, as explained by O. m 0 dx Ocho and N. F. Kolp, ‘‘The computer mouse as data acquisition inter- 2 2 . A4 face,’’ Am. J. Phys. 65, 1115–1118 1997. x 8 k A n Ax dx We used an electronic balance with an accuracy of 0.1 g to measure the weight of ten equal metal washers for a total of 11.2 g. Each time one extra The case of SHM (n1) may be integrated analytically, mass m is displaced from one body to the other, the force changes by the leading to the known result T2m/k, and for the restor- quantity F2mg0.0224 N. 9 ing force Fk sgn X we find T42m/kA. The cylinder top and bottom have spherical shape in order to reduce the oscillation damping. In the case of the twin triangles (n2) and twin cones 10This ‘‘mass trimming’’ is obtained by using small lead spheres those (n3) we may use instead numerical integration, obtaining normally used to load fishing wires that were clamped to the wire above T6.87m/k/A, and T7.42m/k/A, respectively. We the counterweight. 11 used the function ‘‘Nintegrate’’ within the standard An order of magnitude of the hydrodynamic mass mh may be calculated, 16 following K. Thompson ‘‘Hydrodynamic mass,’’ Am. J. Phys. 56, 1043 ‘‘MATHEMATICA’’ software package. 1988, with the simplified assumption of a long cylinder moving end on: Numerical integration may be easily performed also when mh(w /b(2r/L)ln(L/r)1m2 , where w and b are the densities of the restoring force has a polynomial form as for real twin water and brass, respectively. In our case we get a mass correction triangles and twin cones, which cannot have a zero cross (mh /M) of about 2%. section at the vertex. 12This effect can also be detected by a careful inspection of the plot of Fig. 8. The full circles belong alternately to straight lines with different slopes: aAlso at Istituto di Chimica e Tecnologie Inorganiche e Materiali Avanzati those marked ‘‘OUT’’ are the period values calculated from half-periods ICTIMA—Consiglio Nazionale delle Ricerche, Padova, Italy. spent ‘‘out of water’’ an expected smaller slope while those marked 1From now on, the constants k, k, k, , ... will be assumed to be ‘‘IN’’ are calculated from half-periods ‘‘inside water’’ an expected larger positive. slope. 13 2Several experimental setup, have been proposed to investigate the ‘‘x3’’ This can also be detected by a careful eye inspection because the free oscillator with a mass-spring system: e.g., J. Thomchick and J. P. water surface moves up and down by about 1.3 mm. McKelvey, ‘‘Anharmonic vibrations of an ideal Hooke’s law oscillator,’’ 14V. F. Weisskopf and H. Bernstein, ‘‘Search for simplicity: Thermal ex- Am. J. Phys. 46, 40–45 1978; S. Whineray, ‘‘A cube-law air track os- pansion,’’ Am. J. Phys. 53, 1140–1141 1985. cillator,’’ Eur. J. Phys. 12, 90–95 1991; A. Cromer, ‘‘The x3 oscillator,’’ 15See, for example, C. Hirata and D. Thiessen, ‘‘The period of Fkxnx Phys. Teach. 30, 249–250 1992; N. C. Bobillo-Ares and J. Fernandez- harmonic motion,’’ Phys. Teach. 33, 562–564 1995. Nunez, ‘‘Two-dimensional harmonic oscillator on air table,’’ Eur. J. Phys. 16Available for Macintosh and PC-IBM from Wolfram Research at low cost 16, 223–227 1995. in the Education version for teachers and students.
MEMORY LOSS Physics is largely an attitude of mind and I like to think that if I should go to bed tonight and wake up in the morning to find that I had forgotten everything that I had ever learned, but had succeeded in retaining such experience as I have in thinking, I should not have suffered very much by the loss. It would, of course, be a little inconvenient to fail to have ready at hand some of the formulas and methods which are so familiar to us, but this loss could soon be repaired.
W. F. G. Swann, ‘‘The Teaching of Physics,’’ Am. J. Phys. 193, 182–187 1951.
235 Am. J. Phys., Vol. 67, No. 3, March 1999 Pecori, Torzo, and Sconza 235
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